Scientists have trained a machine learning tool to capture the physics of electrons moving across a lattice, using far fewer equations than would normally be required, all without sacrificing accuracy. Whereas until now a daunting quantum problem required 100,000 equations, physicists used artificial intelligence to compress them into a small task of just four equations. All of this was done without sacrificing accuracy.
Illustration of abstract quantum physics
This work could revolutionize the way scientists study systems that contain many interacting electrons. In addition, if it can be extended to other problems, the approach has the potential to help design materials with extremely valuable properties, such as superconductivity or for clean energy generation.
The study, conducted by researchers at the Flatiron Institute and their colleagues, was published in the Sept. 23 issue of Physical Review Letters.
“We started with this huge object of all these differential equations coupled together; then we used machine learning to turn it into an equation so small that you could calculate it with your fingers,” said Domenico Di Sante, lead author of the study. He is an assistant professor at the University of Bologna in Italy and a visiting fellow at the Center for Computational Quantum Physics (CCQ) at the Flatiron Institute in New York City.
The challenging quantum problem involves the behavior of electrons as they move across a lattice-like lattice. When two electrons occupy the same lattice position, they interact with each other. This model, known as the Hubbard model, is idealized for several important classes of materials and allows scientists to understand how the behavior of electrons gives rise to very popular phases of matter, including superconductivity, which is the resistance-free flow of electrons through a material. The model can also serve as a testing ground for new methods before they are released in more complex quantum systems.
The figure shows a visualization of a mathematical apparatus used to capture the physics and behavior of electrons moving on a lattice. Each pixel represents a single interaction between two electrons. Until now, accurately capturing the system required about 100,000 equations – one for each pixel. Using machine learning, scientists have reduced the problem to four equations. This means that a compressed version of a similar visualization requires only four pixels.
However, the Hubbard model is deceptively simple. Even with a small number of electrons and the most advanced computational methods, this problem requires a large amount of computational power. This is because when electrons interact, they become quantum mechanically entangled. This means that even if they are far apart at different lattice points, the two electrons cannot be treated separately. Therefore, physicists need to deal with all the electrons at once, rather than one at a time. As more electrons are added, more entanglement emerges, making this daunting computational challenge exponentially more difficult.
One way to study quantum systems is through the use of so-called renormalization groups. This is a mathematical apparatus that physicists use to study how the behavior of a system – such as the Hubbard model – changes when researchers modify properties such as temperature or observe properties at different scales. Unfortunately, a renormalization group that keeps track of all possible couplings between electrons and doesn’t sacrifice anything can contain tens, hundreds of thousands or even millions of individual equations that need to be solved. On top of that, these equations are quite tricky: each equation represents the interaction of a pair of electrons.
Di Sante and his colleagues wondered if they could use a machine learning tool called a neural network to make renormalization groups more manageable. Neural networks are like a cross between a crazy switchboard operator and the evolution of survival of the fittest. First, a machine learning program creates connections within the full-size renormalization group. Then, the neural network adjusts the strength of these connections until it finds a small set of equations that produces the same solution as the original, large-size renormalization set. The program’s output captures the physics of Hubbard’s model, even though there are only four equations.
It is essentially a machine with the ability to find hidden patterns,” says Di Sante. When we saw the results, we said, ‘Wow, that’s more than we expected.’ We were really able to capture the relevant physics.”
Considerable computing power was required to train the machine learning program, which ran for several full weeks. The good news is that now that their program is optimized, they don’t have to start from scratch after dealing with other problems.
Ultimately, the big open question is how well the new approach works in more complex quantum systems, and the exciting possibilities this technique has when dealing with other areas of renormalization groups, such as cosmology and neuroscience.
